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The aim of this monograph is to give a thorough and self-contained account of functions of (generalized) bounded variation, the methods connected with their study, their relations to other important function classes, and their applications to various problems arising in Fourier analysis and nonlinear analysis. In the first part the basic facts about spaces of functions of bounded variation and related spaces are collected, the main ideas which are useful in studying their properties are presented, and a comparison of their importance and suitability for applications is provided, with a particular emphasis on illustrative examples and counterexamples. The second part is concerned with (sometimes quite surprising) properties of nonlinear composition and superposition operators in such spaces. Moreover, relations with Riemann-Stieltjes integrals, convergence tests for Fourier series, and applications to nonlinear integral equations are discussed. The only prerequisite for understanding this book is a modest background in real analysis, functional analysis, and operator theory. It is addressed to non-specialists who want to get an idea of the development of the theory and its applications in the last decades, as well as a glimpse of the diversity of the directions in which current research is moving. Since the authors try to take into account recent results and state several open problems, this book might also be a fruitful source of inspiration for further research.

Functions of bounded variation. --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables --- Boundary Value Problem. --- Bounded Variation. --- Continuity Properties. --- Fourier Analysis. --- Monotonicity Properties. --- Nonlinear Composition Operators. --- Nonlinear Integral Equation.

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Calculus of variations. --- Differential equations, Partial. --- Functions of bounded variation. --- Mathematical optimization. --- Sobolev spaces. --- Acqui 2006 --- Mathematical optimization --- Calculus of variations --- Sobolev spaces --- Functions of bounded variation --- Differential equations, Partial

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Algebraic topology --- Homology theory. --- Functions of bounded variation. --- Topological spaces. --- Homologie --- Fonctions à variation bornée --- Espaces topologiques --- 51 <082.1> --- Mathematics--Series --- Derived categories (Mathematics) --- Homology theory --- Fonctions à variation bornée --- Functions of bounded variation --- Topological spaces --- Spaces, Topological --- Cohomology theory --- Contrahomology theory --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables --- Abelian categories --- Functional analysis --- Analyse fonctionnelle --- Homologie. --- Groupes topologiques. --- Cohomologie. --- Topological groups --- Algèbre homologique. --- Algebra, Homological

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Calculus of variations. --- Functions of bounded variation. --- 517.57 --- Harmonic functions and their generalizations. Subharmonic functions. Polyharmonic functions. Plurisubharmonic functions --- 517.57 Harmonic functions and their generalizations. Subharmonic functions. Polyharmonic functions. Plurisubharmonic functions --- Calculus of variations --- Functions of bounded variation --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima

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Functions of bounded variation represent an important class of functions. Studying their Fourier transforms is a valuable means of revealing their analytic properties. Moreover, it brings to light new interrelations between these functions and the real Hardy space and, correspondingly, between the Fourier transform and the Hilbert transform. This book is divided into two major parts, the first of which addresses several aspects of the behavior of the Fourier transform of a function of bounded variation in dimension one. In turn, the second part examines the Fourier transforms of multivariate functions with bounded Hardy variation. The results obtained are subsequently applicable to problems in approximation theory, summability of the Fourier series and integrability of trigonometric series. .

Functions of bounded variation. --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables --- Harmonic analysis. --- Abstract Harmonic Analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis